Can I fix broken shape keys after editing basis?
You don't need to delete and recreate the entire shape keys, just fix the points that have changed.

In edit mode, Mesh -> Vertices -> Blend From Shape is useful here.  It copies the positions of selected vertices from one shape key to another, so you can select just the vertices you've changed and copy them to keys where they're broken.  You generally want to turn off the "Add" checkbox when using the Blend, so it copies the positions directly rather than offsetting them.